Does it make a difference if they donate money or objects (e.g. a sefer Torah)? Does it make a difference who the donor is and what their intentions are? Are there cases where donations from non-Jews are accepted?

1 Answer
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Of course they are acceptable if the gentile sincerely wants to donate toward a Torah cause. The Passuk says that a Gentile can bring a Sacrifice to the Beth Hamikdosh see Nazir 62a

איש איש לרבות את העובדי כוכבים שהן נודרים נדרים ונדבות כישראל

We know Herod rebuilt the Beth Hamikdosh through the advice of Bava Ben Buta even though he was an Ever Cnaani (though one may argue he was obligated to do Mitzvos like a woman anyway see Bava Basra 3b-4a).

However it is not our job to expect anything from them or push them to give Bava Basra 4a:

Although tzedaka is similar to sacrifices, they are not the same. See Shulchan Aruch Yoreh Deah 254 about tzedaka from gentiles. The institutions that do accept donations don't have on what to rely on(at least I haven't seen any sources permitting it).
– chacham NisanSep 30 '18 at 11:15

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How do you know Herod was an Eved Knani? I thought he was descended from Edomite converts, and was a full-fledged Jew.
– ezraSep 30 '18 at 14:29